We know that under uniformly continuous function a Cauchy sequence goes to a Cauchy sequence. Let $f:A\rightarrow \mathbb{R}^m$ be uniformly continuous function where $A\subseteq\mathbb{R}^n$, I need to prove $\lim_{n\rightarrow\infty}f(x_n)$ exists where $x_n$ is a Cauchy sequence in $A$, well If I write coordinate wise of the element of $f(x_n)$ then coordinatewise each sequence is Cauchy and hence they will converge to some element in $\mathbb{R}^m$